13. Appendix

Computational engineering, especially linear algebra, has an abundance of material beyond what is reasonable to cover in this short and applied introduction. Included here are some material including explanations of useful algorithms, definitions, properties of certain matrices, and proofs that beg for some coverage.

Students are not accountable to know the material in the appendix for exams.

• 13.1. Norms
  • 13.1.1. Vector Norms
  • 13.1.2. vecnorm function
  • 13.1.3. Matrix Norms
• 13.2. Vector Spaces
  • 13.2.1. Vector Space Definitions
  • 13.2.2. Linearly Independent Vectors
  • 13.2.3. Rank
• 13.3. Finding Orthogonal Basis Vectors
  • 13.3.1. The Gram–Schmidt Algorithm
  • 13.3.2. Implementation of Classic Gram–Schmidt
  • 13.3.3. Implementation of Modified Gram–Schmidt
• 13.4. Linearly Independent Eigenvectors
  • 13.4.1. Pairwise Independence
  • 13.4.2. General Independence
• 13.5. All About the Number e
  • 13.5.1. Definition and Derivative
  • 13.5.2. Euler’s Complex Exponential Equation
  • 13.5.3. Numerical Verification of Euler’s Formula
  • 13.5.4. Compound Interest
• 13.6. A Matrix Exponent and Systems of ODEs
  • 13.6.1. A Matrix in the Exponent
  • 13.6.2. Example Matrix Exponent
  • 13.6.3. Matrix Solution to a System of ODEs
  • 13.6.4. Example ODE Matrix Solution